Questions tagged [extension]
The extension tag has no summary.
65 questions
12 votes
0 answers
384 views
"Non-Abelian" "extensions" of Lie algebras
Introduction In traditional Lie algebra cohomology, one is able to classify extensions of a very specific type — given the following information: Lie algebra $\mathfrak g$ A $\mathfrak g$-module $M$ ...
- 509
3 votes
1 answer
258 views
Extensions of diagonalizable, respectively multiplicative-type, groups
In [Milne, Example 12.10], the author states: "Later (12.22, 15.39) we shall see that an extension of diagonalizable groups is diagonalizable if if it is commutative, which is always the case if ...
- 14k
3 votes
0 answers
171 views
The relation of classifying stacks and central extensions
Let $G$ and $E$ be 0-truncated group objects in the infinity category of stacks on a Grothendieck site. Suppose $E$ is commutative. Then it turns out that the classifying stack $BE$ of $E$-torsors ...
- 526
1 vote
0 answers
95 views
Analyzing subquotients of a representation
Let $G$ be an abstract group and let $V$ be a finite dimensional irrep over $\mathbb{C}$. Let $E$ be an extension of $V$ by itself, and consider $E^2 := E\otimes E$, the tensor square of $E$. $E^2$ ...
- 3,346
2 votes
0 answers
86 views
Subquotients of tensor products of extensions
Let $G$ be an affine group scheme and let $\mathcal{T}$ denote the Tannakian category of its representations. Let $V$ be a semisimple object of $\mathcal{T}$. Fix $\mathcal{T}_0$ to be the full ...
- 3,346
0 votes
1 answer
182 views
A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
- 1,643
3 votes
1 answer
320 views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 4,942
3 votes
1 answer
325 views
Extension of Sobolev function defined on unit cube
Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
- 103
1 vote
1 answer
161 views
A correspondence between projective representations of $G$ with those of its universal cover
Let $G$ be a connected Lie group and $\mathcal{H}$ be a Hilbert space. Let $U(\mathcal{H})$ denote the the group of all unitary operators on $\mathcal{H}$ with function composition (i.e., $\hat{U}:\...
- 297
2 votes
2 answers
255 views
Extensions of $G$-modules parametrized by $H^1$
Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
- 339
1 vote
0 answers
110 views
How can we construct a non-trivial central extension of a Lie group
Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
- 297
5 votes
1 answer
225 views
Explicit formula for general group extension in terms of cartesian product set
According to Wikipedia and ncat lab general group extensions $$N\rightarrow G\rightarrow Q$$ are classified by a group homomorphism $$\rho: Q\rightarrow \operatorname{Out}(N)$$ subject to a constraint ...
- 3,095
6 votes
2 answers
780 views
How is the classification of groups extensions by $H^2$ related to Yoneda Ext?
It is well-known that group extensions $$1\to A \to H \to G \to 1$$ where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
- 4,077
1 vote
1 answer
247 views
Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$
Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
- 409
2 votes
0 answers
213 views
Homotopy equivalence of chain complexes from subcomplexes and quotient complexes
Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
- 713
5 votes
1 answer
220 views
On the property P in the Whitney extension theorem
Let $D$ be a possibly unbounded domain in $\mathbb{R}^d$, $d \ge 2.$ We say that $D$ has the property P if there exists $C>0$ such that such that any pair of points $x,y \in D$ can be joined by a ...
- 807
1 vote
0 answers
65 views
extension from a dense subset in completely uniformizable spaces
Let $\mathbf{CReg}$ the category of completely regular spaces and continuous maps and let $\mathbf{Unif}$ be the category of uniform spaces and uniformly continuous maps. There is a functor $F:\mathbf{...
3 votes
1 answer
235 views
Homeomorphic extension of a discrete function
Let $f : \{ 0,1 \} ^ {n} \rightarrow \{ 0,1 \} ^ {n}$ be a bijective map. Then is there a known computable way to extend it to a homeomorphism $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
- 31
2 votes
1 answer
246 views
Lie algebras for which all one-dimensional extensions split
I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
- 1,367
4 votes
2 answers
197 views
A ball with slit at the radius is not $W^{1,1}$-extension domain
Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
- 1,155
3 votes
1 answer
185 views
Boundedness of an extension operator
Let $d \ge 2$ be a positive integer. For $x=(x_1,\dotsc,x_{d-1},x_d)$, we write $x'=(x_1,\dotsc,x_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x_d) \mid x_d>0\}$ denote the $d$-dimensional upper half-space. ...
- 807
2 votes
0 answers
134 views
Any connection between extension of algebraic structure and forcing of set theory?
Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
- 3,296
5 votes
1 answer
340 views
Extension of first order deformations of a line bundle
Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
1 vote
0 answers
70 views
$H^1 \cap C^0$ boundary, smooth $H^1$ extension
Assume we have a $u \in H^1(\Omega; \mathbb{R}^n) \cap C^0$ where $\Omega$ is a bounded open Set with smooth boundary. Also $u\vert_{\partial \Omega} \in H^1(\partial \Omega; \mathbb{R}^n) \cap C^0$. ...
- 11
2 votes
0 answers
126 views
Reference for an extension theorem for Neumann boundary data
$\DeclareMathOperator\Tr{Tr}$Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain (we denote by $n$ the normal to $\partial\Omega$) and $p\in(1,\infty)$. Do you know where I can find (book or ...
- 499
8 votes
3 answers
826 views
Is there some example that nicely extends the multiplication of natural numbers?
Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
- 181
4 votes
1 answer
772 views
Yoneda Ext theorem and extensions
Consider the category of chain complexes over a ring $R$. We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \...
user30211
2 votes
1 answer
529 views
$C^1$ extension with compact support
Knowing that $\omega\Subset\Omega\subset\mathbb{R}^2$ (compactly included) are two open and bounded sets with $C^2$ boundary, is it true that for any function $\phi_0:\overline{\omega}\to\mathbb{R},\ \...
- 2,029
2 votes
0 answers
152 views
Extensions in a full subcategory
Let $\mathcal{C}$ be an abelian category (feel free to put more adjectives here) and $\mathcal{D}$ a full abelian subcategory closed under kernels and cokernels. Then by definition for $A,B\in \...
- 397
4 votes
0 answers
175 views
Continuous extension preserving modulus of continuity
Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
2 votes
0 answers
301 views
Extension of Lipschitz functions that preserve the Frobenius norm of the Jacobian
Let $n,m\ge 1$ be integers and let $f:E\to R^m$ be $L$-Lipschitz for some subset $E\subset R^n$. Kirszbraun's theorem, https://en.wikipedia.org/wiki/Kirszbraun_theorem, states that there exists ...
- 1,834
2 votes
0 answers
67 views
Extension of differentiable structure to guarantee continuous extension of prescribed vector fields
Consider the lower half space $\{(x,y) \in \mathbb{R}^2 \; | \; y < 0\}$ and let $X_1, X_2$ be two vector fields on the lower half space which are continuous with respect to its canonical ...
- 413
6 votes
0 answers
288 views
How much does Ext tell me about isomorphisms?
So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
- 69
1 vote
0 answers
62 views
Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?
Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$. Let $...
- 117
6 votes
1 answer
221 views
Extension Operator for $W^{1,\infty}(U,X)$
I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
1 vote
1 answer
151 views
Computation of extension groups in the category of pairs $(M,f)$
Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear ...
- 1,573
2 votes
1 answer
252 views
Homomorphisms of ring extending nicely ideal intersections
Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$. Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$. Is there some "natural" assumption on $\varphi$ to ...
- 43
1 vote
1 answer
132 views
Extend a bundle "trivially"
Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with $E'|...
- 1,653
21 votes
1 answer
880 views
Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure
Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...
- 14.5k
1 vote
0 answers
161 views
Finite field extension
Suppose $$f_a=x^p-x-\left [\prod_{j=1}^{a-1} \alpha_j\right]^{p-1} \in \mathbb{F}_p(\alpha_1,\ldots,\alpha_{a-1})[x]$$ is irreducible over $\mathbb{F}_p(\alpha_1,\ldots,\alpha_{a-1}) $, where $\...
- 111
5 votes
0 answers
286 views
Do all fields with internal absolute values arise as ordered fields or like $\mathbb{C}$ from them?
$\def\abs#1{\lvert#1\rvert} \def\Im{\operatorname{Im}} \def\Re{\operatorname{Re}}$ (Crossposted from math.stackexchange.com after 5 days with no correct answer.) Let $\langle F,+,\cdot\rangle$ be ...
user5810
5 votes
2 answers
3k views
Extensions of local vector fields to whole manifold
Let $M$ be a smooth manifold (with boundary). Suppose I have a smooth vector field $T$ defined on the complement of a compact subset $K$ of $M$ and I wish to extend $T$ to the whole of $M$. What are ...
- 914
2 votes
0 answers
82 views
Ref Request: Extension Operators for Slobodeckii Spaces of Higher Order
I have been looking for (linear) Extension Operators for Slobodeckii spaces $W^{s,p}(\Omega)$ where $s>1$ and $\Omega \subset\mathbb{R}^N$ is a sufficiently smooth domain, where the influence of $\...
- 351
4 votes
1 answer
504 views
busby invariant of extensions of $C^*$-algebras
I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras". Let $0\to B\to E\to A\to 0$ be a short exact ...
user62639
1 vote
0 answers
492 views
Quotient of two smooth functions extension
Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...
- 501
6 votes
1 answer
419 views
Extension of functions from geodesically convex compact sets in a Riemannian manifold
In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
- 36.6k
9 votes
0 answers
147 views
General approaches to extension theorems as Caratheodory
I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear. I ...
3 votes
1 answer
3k views
Extension of continuous and smooth functions
Let us consider any subset $U \subset \mathbb{R}^{n}$. By definition, a function $f: U \rightarrow \mathbb{R}^m$ is smooth if, for every $x \in U$, there exist an open neighbourhood $\Omega_{x}$ of $x$...
- 1,282
3 votes
1 answer
1k views
Fractional Sobolev spaces and extension by zero
The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero to whole $\mathbb{R}^n$ (...
- 51
5 votes
2 answers
509 views
Conjugation in associative algebras over finite fields
Let $A$ be a finite dimensional associative algebra (with unity) over a finite field $F$. Let $L$ be a field extension of $F$. Suppose that after extending scalars to $L$, two elements $a,b$ of $A$ ...
- 367